I'm not sure where to go from here in my uv mapping!
That's kinda weird looking (the shading artifacts, not the model). Maybe you have some extra faces in the model you need to delete?

Assuming that's not the case, first remember that lamp shades don't have tops (unless you want this one to), so you can delete that face if you like.

If you still need them, they will cause you problems if you have faces with more than 4 sides. The fix for that is to select each n-gon ("circular"-looking face with many sides) in the modeling view and go to face -> poke faces. This will create a triangle fan pattern but leave your current UV's intact.

Now, a feature of UV mapping you might not be aware of is called "seams", as in seams on pieces of cloth that get stitched together to make clothes. In the modeling view, select all the edges that indicate separate pieces of the model (those would be the edge loops that separate the top of the lampshade from the cylindrical side, and the cylindrical side from the bottom. With those selected do Edges -> Mark Seams. You will get red highlights on those edges.  You will also need to select an edge that runs down the lampshade so it can be folded flat, like a cylinder being unfolded on a table. Choose Edges -> Mark seams for that too.

Now choose UV --> Unwrap and your model should be almost perfectly unwrapped for you without any manual adjustments needed.